documentclass[avery5388,grid,frame]{flashcards}

cardfrontstyle[largeslshape]{headings}
cardbackstyle{empty}

%The notable stuff starts here
usepackage{tikz}
usepackage{background}
usetikzlibrary{patterns}
backgroundsetup{%
  opacity=.2,    %% Play with this to increase/decrease readability
  contents={begin{tikzpicture}[remember picture,overlay]
          fill[pattern = crosshatch] (-50,-50) rectangle (50,50);    %% yshift and xshift for example only
    end{tikzpicture}}
}
%%%%%%      And ends here


begin{document}

cardfrontfoot{Functional Analysis}

begin{flashcard}[Definition]{Norm on a Linear Space \ Normed Space}
    A real-valued function $||x||$ defined on a linear space $X$, where15$x in X$, is said to be a emph{norm on} $X$ if
smallskip
begin{description}
item [Positivity]            $||x|| geq 0$,
item [Triangle Inequality]   $||x+y|| leq ||x|| + ||y||$,
item [Homogeneity]           $||alpha x|| = |alpha| :  ||x||$,
$alpha$ an arbitrary scalar,
item [Positive Definiteness] $||x|| = 0$ if and only if $x=0$,
end{description}
smallskip
$x$ and $y$ are arbitrary points in $X$.
medskip
linear/vector space with a norm is called a emph{normed space}.
end{flashcard}

begin{flashcard}[Definition]{Inner Product}
$X$ be a complex linear space. An emph{inner product} on $X$ is
a mapping that associates to each pair of vectors $x$, $y$ a scalar,
denoted $(x,y)$, that satisfies the following properties:
medskip
begin{description}
item [Additivity]            $(x+y,z) = (x,z) + (y,z)$,
item [Homogeneity]           $(alpha : x, y) = alpha (x,y)$,
item [Symmetry]              $(x,y) = overline{(y,x)}$,
item [Positive Definiteness] $(x,x) > 0$, when $xneq0$.
end{description}
end{flashcard}

begin{flashcard}[Definition]{Linear Transformation/Operator}
    Atransformation $L$ of (operator on) a linear space $X$ into a linear
    space $Y$, where $X$ and $Y$ have the same scalar field, is said to be
    a emph{linear transformation (operator)} if
    medskip
    begin{enumerate}
        item $L(alpha x) = alpha L(x), forall xin X$ and $forall$
            scalars $alpha$, and
        item $L(x_1 + x_2) = L(x_1) + L(x_2)$ for all $x_1,x_2 in X$.5
end{enumerate}
end{flashcard}
end{document}